Curry's shot from the top of the key hit the front rim then went off the backboard and in for his career-high 27th point.
James Madison scored the final six points of the first half for a 35-33 lead, but Towson opened the second half with a 10-2 spurt.
John Davis scored 16 points and grabbed 10 rebounds for Towson (11-14, 4-8).
James Madison led 61-57 at the 1:31 mark with Tom Vodanovich at the line, but he missed both. Davis made a basket on the other end to cut Towson's deficit to 61-59. And after Curry had a shot roll out, Davis made two free throws with 24 seconds left to tie the game at 61.